Joan Richards (1988). Mathematical visions. The pursuit of geometry in Victorian England.

Annotated by Ben Wilbrink

Joan Richards (1988). Mathematical visions. The pursuit of geometry in Victorian England. Academic Press.

The title does not adequately disclose its contents: what is mathematics, what is it about, what is it to be a mathematician, why mathematics in education, how to assess mathematics achievement in education? The Cambridge Mathematical Tripos is the case that is used throughout the book, making it a history of the Mathematial Tripos in the 19th and early 20th centuries.

Yes, Joan Richards here touches on almost every intriguing question that one may pose on mathematics, mathematics in education, and testing mathematics achievement in education.

The mathematics concerned is almost exclusively geometrics, because of the particular circumstance of Euclid's Elements being used as the textbook in secundary education nationwide until approximately 1905, and the (first part of) the Mathematical Tripos demanding proofs exclusively in Euclidean style.

Joan Richards takes her readers on a really fantastic journey, every which way you might look at her books contents. The exposition is rather difficult to follow, however, because of the highly artificial character of the geometries (the non-euclidean and projective ones) involved; but then that is exactly one of the issues the book is about: why bother students with those artificialities, and why bother them with Euclid to begin with? Questions that have a familiar contemporary ring, isn't it?

I. Bernard Cohen in his preface summarizes the major themes, all of them in my opinion directly touching on contemporary issues in mathematics education and assessment: "She [Joan Richards] stresses the great changes thatr occurred in the actual practice of mathematicians as a result of the 'social and philosophical reorganization of knowledge.' She sees the establishment of 'a recognized and viable mathematical resarch community in England' as a force that would change the view of the subject itself. No longer was mathematics to be valued primarily (if not exclusively) for its role in the education of young men and women and its presumed exemplification of the highest form of rationality. The new 'ideology' was that of a research community, in which a 'mathematician' would be valued not for his ideas oncerning education at large or for his ideals of the role of mathematics in education, not even for his gifts of prowess as a teacher or presenter of the subject at a textbook level. Rather, there was a new career in which the major reward would come from recognition for creative additions to the methods and subject matter of mathematics itself."

queen of sciences

"Most nineteenth-century philosophers perceived Euclidean geometrical axioms as descriptions of the fundamental properties of spatial reality."
"Triumphantly accurate in both the subjective and the objective realms, geometry was the summum bonum of human knowledge. It was truly the queen of sciences."

p. 1., 2.

What is it to have mathematical knowledge:

Meno's paradox?

"The extraordinary success of geometry forced all who tried to formulate a theory of human learning to confront the issue of how people could come to have this kind of exact knowledge of absolute truth."

p. 2.

Epistemological theories of Kant, Locke, etcetera. Two positions: nativism versus empiricism.

space produced indefinitely

"Empiricists were hard-pressed to explain how we could conclusively prove the truth of statements about space when proving even the simplest physical statements was not only laborious but inexact as well."
"An empirical theory of knowledge would suggest that valid knowledge is bounded by experience, and yet we have no experience of lines produced indefinitely in the far reaches of space. Euclid's definition of parallel lines asserted knowledge of how these lines would behave, however."

p. 3.

The parallel lines dilemma would eventually lead to highly artificial (formal) non-Euclidean geometries, and projective geometry with its imaginary points in infinity (and therefore not really solving this particular difficulty). What is highly illumating for the sake of mathematical education is the debate over all these developments, however.

The perceptive reader will recognize may points of similarity with the contemporary debate on constructivism and RME. Not knowing ones history doomes one to repeat it. No, this is not Joan Richards' opinion. All comments and annotations are mine, quotations are directly from the book.

non-Euclidean geometries:

"Their cogency undermined the conviction that Euclidean geometry was the definitive presentation of a completely known form of knowledge."

p. 4.

paradigm shift

"Whereas in the nineteenth century, geometrical results were perceived descriptively, as binding truths about real space, in the twentieth they are more commonly seen formally, as deductions drawn from an abstract axiom system only more or less appicable to the real world."

p. 4

A bombshell for the educational enterprise! What used to be regarded as rather concrete, because of its ascribed descriptive character, turned into an abstract system. As Richards will show later in the book, this resulted in mathematicians and educators each going their separate ways at the end of the 19th, the beginning of the 20th century.

The paradigm shift is not an idea of Joan Richardson. However, she refers Thomas Tymoczko (1985). New directions in the philosophy of mathematics, Birkhauser, on whether or not there are paradigm shifts in mathematics also.

descriptive interpretation

"... the circles which are the focus for mathematical discourse are intimately tied to the circles encountered elsewhere - whether as physical, visual or intellectual onbects. In this view, a circle is a circle no matter how Euclid or any other geometer might wish to speak of it; in the final analysis, their mathematics is right or wrong depending on how accurately their circles accord with the circles which are encountered in other contexts."

p. 4-5

the formal view

"In contrast to the descriptive view, from the formal perspective the rlationship between mathematically constructed systems and non-mathematical experience, be it physical, visual or conceptual, is irrelevant to mathematical truth."

To me it is absolutely evident that the formal vision does not touch on the mathematics in primary or secondary education, no way. If traces of formalism figure in contemporary text books or practice, such will make them highly suspicious. A case in point might be the use of set theory in primary education. For the time being this is my working hypothesis, however. Be aware that this position does not in itself entail constructivism of RME, the latter might just be a form of descriptive extremism.


"The aim is to trace the changing interpretations of geometrical truth in relation to a broader cultural context rather than to criticize, evaluate or interpret the ideas being propounded."

For example: "Pure mathematics, even at its highest levels, was more often defended as intellectual training than as an autonomously valuable specialized study."

p. 6, 7.

What is mathematics?

"This question is at once simple and immense. It is not relevant only to the systems of philosophers or even of mathematicians. It is addressed over and over again as societies negotiate the ters of mathematical education, development, support and prestige."

p. 11

That is exactly why this historical study nevertheless touches directly on contemporary discussions about mathematics in education.

In 1818 a translation of a French calculus text appeared, published by Charles Babbage, John Herschel and George Peacock.

"Within a year, the format of the Cambridge mathematical examination, also called the Senate House Examination or the Tripos,was changed to allow for the use of non-Newtonian symbology in solving problems.

p. 13.

Many more examples of the interplay between this examination, mathematical developments proper, and the profession of mathematics are to follow. The influence of the Tripos is shattering, really, only to be changed for a more proper position in the early 20th century. An important change had already taken place in 1819, when "Newton's fluxional notation was abandoned on the Senate house Examination. In the ensuing years this innovation took hold, and analytical notations were universally adopted. Abandoning the geomatrical, Newtonian notation of the previous century was advocated as a significant step towards bringing English scientists into the same universe of discourse as their continental counterparts." (p. 16; references given there in note 6 )

"In the wake of the liberating reforms the purview of the examination expanded unmanageably. Since any aspect of mathematics might be included on the Tripos, it became virtually impossible to prepare for it. In the middle of the century, as part of an attempt to define and delimit the subjects appropriate to the examination, the place of mathematics in the Cambridge curriculum was searchingly reexamined and reevaluated."

p. 17

one step forward, one step backward

"In 1818, the members of the Analytical Society had advocated the study of analysis in an attempt to move away from the geometrical emphasis of the Newtonian legacy. This position was reversed in 1848 when the same men were important actors in a move to reemphasize geometry as the core of the Cambridge liberal education.

p. 17.

That geometry was a strictly Euclidian geometry.

mathematics and classics at Oxbridge

The unique aspect of the education at Cambridge, which set if off from Oxford, was that the major emphasis of its curriculum was upon mathematics. Until the 1850's, no matter what subject formed a student's primary interest, he had to study mathematics to obtain an honors degree. At Oxford, the primary focus of the education was on the classics. But at Cambridge, even if Greek and Latin were his major interest, a student could not take the classics examination without first passing the Mathematical Tripos."

p. 19.

Take your time to let the above quotation sink in. Especially to continental European ears this is so out-landish. If you are not already somewhat familiar with the Mathematical Tripos, read the quotation again, and let it sink in again. This examination really is unheard of in the rest of this world, even if highly competitive examinations were already known in Leuven four centuries earlier (see my 1997 html).

Whewell (1835 pamphlet)

"Let us suppose it established, therefore, that it is a proper object of education to develop and cultivate the reasoning faculty. The question then arises, by what means this can be done; - what is the best instrument for educating men in reasoning"

Richards: "The answer Whewell offered was mathematics. It was through mathematical study, he claimed, that students could learn to reason effectively in all areas of their adult lives. (...) It was because of its power to train the faculty of reason rather than its value as a useful study per se that Whewell argued for the study of mathematics. (...) In Whewell's view, a mathematically educated person would make a good lawyer, parliamentarian, doctor or clergyman because his studies would have tought him how to think effectively."

p. 21.

This kind of claim was made on a regular basis by many writing on education in the 19th century (and the centuries before that; faculty psychology, of course, was an early 19th century phenomenon); nobody ever thought of this kind of claim as an empirical one, that might be tested by doing appropriate research, and might thus be refuted. Contemporary politicians and columnists still are in the habit of claiming such things, unaware of the existence of relevant theory and research. Hans Freudenthal has written in this vein, also; not all of his work, still some of his work. Hamilton, a Scottish hilosopher, did not agree with Whewell, however. "In Hamilton's view, then, mathematics was different from other studies because it was formally based. As such it was an empty and isolated study; hence it had virtually no value as a part of the liberal education." (p. 22). Hamilton did not need empirical data at all. As described, Hamilton's view entails the disappearance of mathematics altogether, being no good at all for anything or anybody outside of mathematics. In fact, that is what a committee in 1852 feared would happen to mathematics if was not forcefully promoted in the way Cambridge did in its Mathematical Tripos (p. 19). Wonderful times, the 19th century.

necessary and contingent truth

"Whewell illustrated the distinction between necessary and contingent truth with kepler's and Newton's work on elliptical planetary orbits. When Kepler concluded that planets rotated in elliptical orbits, it was imaginable to him that they might have followed a different path; it was only his observational data which led him to this conclusion. In Whewell's mind, this kind of understanding of planetary orbits was in marked contrast to Newton's. Because he understood the nature of force, Newton knew not only that the orbits were elliptical but also that they could be nothing but elliptical. In his clearly conceived concept of force, Newton's mind was in essential harmony with the external world. Whewell maintained that whereas Kepler had developed only a contingent description, Newton knew the form or planetary orbits necessarily."

p. 28-29

In the meantime we have learned that this kind of necessary truth nevertheless is relative to a proposed theory, be it that of Newton, Einstein, or whatever. The difference in perception of truth is far from trivial, however.

necessary and conservative truth

"Whewell's category of necessary truth was critically important for the assurance that man really could come to know his world. This assurance in turn supported his basically conservative outlook in which there were certain immutable truths about God, man and society which the educated elite, of which he was a member, understood and passed down from generation to generation. His category of necessary truth was not only relevant to abstract epistemological arguments, but in the 1840's, buttressed a much broader and highly conservative political, social and theological outlook."

p. 29

Whewell was quite fundamentalistic about 'mathematical truth', Richards sumamrizes this to 'the human mind was completely in tune with external mathematical fact.'

merely experience

"If any person does not fully apprehend, at first, the different kinds of truth thus pointed out [contingent and necessary], let him study, to soem extent, those sciences which have the necessary truth for their subject, as geometry [the study of space], or the properties of numbers [arithmetic], so as to obtain a familiar acqaintance with such truth; and he will then hardly fail to see how different the evidence of the propositions which which occur in these sciences, is from the evidence of the facts which are merely learnt from experience."

[quotation from Whewell (1947), Philosophy p. 58] p. 29

Whewell's outlook of life was a cultural thing, of course. Anybody proclaiming such a thing in the 21st century must be crazy or a CEO. In his days, his nativism was opposed by Herschel's empiricism, an empiricism mixed up with theology, however.

Herschel's mathematics: no formalism

"To be significant, the truth of mathematical statements had to lie in the subject matter being described; it could not be confined to empty manipulations of terms and figures which were merely subjectively defined.
Thus, by the middle of the century, although apparently poles apart philosophically, Herschel and Whewell seem to have been close together in the essential outlines of their views of mathematics. Both of them insisted that significant mathematics was essentially descriptive and argued vehemently against suggestions that it might be otherwise conceived."

p. 33-34

Both visions, Whewell's and Herschell's, would be shattered by developments to come: non-Euclidean and projective geometries. Would that pendulum swing back again? Yes, because by the end of the century these new geometries were seen to be of no value at all in describing worldly/scientific phenomena. Mathematics and science would each go its own way.

Then the Tripos. In Peter Searby's (1997) A history of the University of Cambridge, Volume III, 1750-1870 (Cambridge University Press), a description of the Mathematical Tripos is given on p, 176-186. It is very difficult to get from his description a clear picture of the examination. One reason for this to be so, is the utter strangeness of this examination in contemporary eyes. Another might be that Searby is not really interested in things mathematical, he does not even mention the work of Joan Richards, for example. Therefore I will quote here the clear text of Joan Richards on the characteristics of the Mathematical Tripos in 1848, before and after that crucial date that means. There are some differences of detail between the desciptions given by John Searly and Joan Richards, but I will not go into them.

The Tripos in the revolutionary year 1848

"Before 1848, the Tripos was an undifferentiated six-day examination. In the reform of 1848 it was lenghtened to eight days and divided into two pats. The first three days were designed to cover the material essential for anyone to receive an ordinary degree. Only after he had completed the first part of the examination could a student sit for the more advanced, second part of the examination. His performance on this second part determined whether he would receive honors. Until 1851 students had to receive mathematical honors before they were allowed to compete for honors on the Classical Tripos. After that date, when the Moral Sciences Tripos and Natural Sciences Tripos were added, students could attempt to receive honors on any Tripos after taking only the first part of the Mathematical Tripos. Thus, until the end of the century, the first part of the Mathematical Tripos remained the solid core of the education of any Cambridge graduate. This meant that when they fixed the form of the first part of the Tripos, the reformers were deciding on those subjects which would contribute to the backbone of the Cambridge education. The issues surrounding this decision illustrate the kinds of implications attendant on embedding mathematics into the Cambridge liberal education and more generally into English intellectual culture as a whole."

p. 40

contents of the first part of the Mathematical Tripos

"... the portions of Euclid usually read; Arithmetic; parts of Algebra, embracing the Binomial Theorem and the Principles of Logarithms; Plane Trigonometry, so far as to include the solution of Triangles; Conic Sections, treated geometrically; the elementary parts of Statics and Dynamics, treated without the Differential Calculus; the First three Sections of Newton, the Propositions to be proved in Newton's manner; the elementary parts of Hydrostatics, without the Differential Calculus; the simpler propositions of Optics, treated geometrically; the parts of Astronomy required for the explanation of the more simple phenomena, without calculation."

Cambridge University Commission (1852) Appendix to Evidence from the University, p. 232. As cited on p. 40-41.

"Each of these specifications reflects a descriptive view of mathematics. They require that diverse mathematical subjects be taught by geometrical methods, even when more efficient analytical ones were readily available."

p. 41.

This emphasis of geometry was a reversal of an earlier trend to emphasize analysis; one of the architexts of this reversal was William Whewell. .

D. A. Winstanley (1940). Early Victorian Cambridge. Cambridge University Press.

Analysis symbolizes content away

"Whewell here [a quotation I have omitted here, b.w.] criticized analysis because the power of its symbology obscured the focus on a particular subject matter which was essential to valid science. For this reason, geometry, which required that one concentrate directly on the known properties of space in order to solve particular problems, served as significantly better training."

p. 42.

I just might be the case that educators in the nineteenth century still were aware of certain characteristics of analysis (including algebra, of course) that made it less fit as a subject to be treated on its own. What has changed in the one and a half century since: this character of analysis, or the insights of educators? And if the last option is the true one, is that a change for the better or the worse, and what direction do you think the change has taken us? Maybe old man Whewell hit this nail right on the head. Fascinating. I'd like to see contemporary educators comment on precisely this point, arguing on the basis of sound theory and experiment, of course.

young Herschel's opposing view

"Generality was the mark of scientific power, and analogies which could be found to hold among many disparate phenomena bore with them the characteristics of vera causa and the stamp of truth. In this viw, the flexible generality of analysis was closer to creative scientifi thinking than geometry, and therefore more valuable."

p. 42.

What is the value of algebra in the secundary curriculum? Contemporary educators seem to view it the way Herschel did. One of the problems is that such a view does not automatically result in well designed instructinal practices. Quite to the contrary, I would say. Yet contemporary educators suggest that they know perfectly well the good courses from the bad ones, most contemporary courses belonging to the last category, of course. What is 'good': a lot of content and practice. But such a didactical theory is known as the bucket theory: knowledge is there to be presented to and digested by the pupils, as simple as that.

Augustus de Morgan's insistence on interpretation

"No science of symbols can be fully presented to the mind, in such a state as to demand assent or dissent, until its peculiar symols, their meanings, and the rules of operation, are all stated." In his 1849 Trigonometry and Double Algebra, p. 89.

"De Morgan's insistence that interpretation was the culmination of any complete algebraic investigation guaranteed that algebra would, like geometry, be a descriptive study. From this perspective, legitimate algebra must always be accompanied by a subject matter through which it could be interpreted.
De Morgan's perspective, which allowed for the value of symbolic manipulation but insisted that ultimately mathematical truth was to be found in its interpretation, succeeded in saving analytic arguments from being totally discredited within the mid-century English view of mathematics."

p. 46-47.

a real understanding

"The subjects for matematical interpretation could be drawn from a wide range of physical investigations, but interpretations were always necessary for a real undertanding. Ultimately mathematics had to remain descriptive despite all temptations to move in a more formal direction."

p. 49-50.

Thus was the opinion of the already mentioned 1952 Commission report, in line with a balanced testimony by the young Robert Ellis, founder of The Cambridge Mathematical Journal (Richards, p. 47 ff.).

Imaginary (complex) numbers posed difficulties for those trying to interpret them geometrically. Geometrical truth came under pressure. Transcendental spaces of more than three dimensions were proposed. Who could 'imagine' those spaces? Helmholtz couldn't, and many with him. Non-Euclidean and projective geometries would soon lead to more suffering for believers in geometrical 'truth' (Richards chapter 2). I will more or less skip the debates about geometrical truth in chapter 2. The non-Euclidean aspect of these alternatives is in the treatment of parallels only.

Euclid's fifth postulate and definition of parallel lines

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

Parallel straight lines are straight lines which, being in the same plane and being produced idefinitely in both directions, do not meet another in either direction."

From Thomas L. Heath (1956) The Thirteen Books of Euclid's Elements. pp. 154-55. Quotation on p. 62.

The experienced designer of achievement test items immediately will recognize the term 'indefinitely' as the potential trouble maker. And it is. Along comes Saccheri, assuming the postulate is false, and proving and then trying "to demonstrate the falsity of any alternative postulate." (p. 63). Which he managed to do. Saccheri already published his work in 1733.

Then Lobachevskii and Bolyai came along, almost a century later. And so it went. Retinal images played their part also in Berkely' and reid's philosophies. Riemann wrote his Habilitationsvortrg, translated by Helmholts in English. The flatland metaphore was invented, by Helmholtz I think Richards, p. 96 ff.), as an attempt to 'imagine' what was not imaginable. Edwin Abbott romanticized it in his 1884 Flatland. Dutch readers visit platland

truth mathematical and theological

"Within the particular configuration of knowledge the nineteenth-century English had constructed, mathematical truth was central to theology. The transcendental truth mathematics was believed to describe had long stood as the exemplar of the perfect truth to whih the human intellect aspired. This was a particularly central issue in theology, where it was often argued that knowledge of the divine partook of the same transcentdental necessity as knowledge of mathematics. In this tradition, knowledge of god was defended by being equated with the unquestioned status of geometrical truth." (see also Imre Toth, 1982, Gott und Geometrie: Eine viktorianische Kontroverse, in Dieter Henrich: Evolutionstheorie und ihre Evolution. pp. 141-204)

p. 104.

This is an explosive relation, as soon as doubt arises about geometrical 'truth.' This is reminiscent of another major crisis in theological belief: the discovery of the Solar system being inherently unstable (Stephen Toulmin, 2001, Return to reason. Harvard University Press).

And then there was also this theory of evolution .... .

"This theological context made mathematical discussions particularly controversial."

p. 104

In the Middle Ages also logicians had to be careful about matters theological.

a new kind of truth: evolution theory

"Darwin's On the Origin of Species, published in 1859, raised a number of serious question [sic] for English views of humanly knowable truth. The Origin severely threatened the tradition of natural theology, not only because of the bloodthirsty nature Darwin portrayed there, but also because of the kind of truth he implied was to be attained through science. Although he apparently had spent twenty years trying to fit his theory into some form of Herschel's and Whewell's framework for legimate science, Darwin's work did not fit the model of scientific truth they had constructed. His Origin contained a highly persuasive argument, but, by traditional standards, Darwin did not seem to have proved the truth of his theory."

p. 104-105

Wow. Situated truth, if ever there was such a truth. A contradiction in terms, of course, but the discussion revealed 'truth' to be subject to opinion. (Richards refers David L. Hull (1973) Darwin and his critics, University of Chicago Press.).

Thomas Henry Huxley and John Tyndall developed a scientific naturalism: knowledge results from observation.

contingent knowledge vs transcendental truth

"They [Huxley, Tyndall and others] firmly claimed that people ccould only claim to know the information reveived through the senses. Such knowldge was always contingent and limited by the the field of possible experience. Transcendental realities, the scientific naturalists insisted, were unknown and unknowable in any field."

p. 105.

Richards then goes on to report on the discussion on cause, such as the statement that gravity causes rocks to fall. A beautiful case to use in education, for example education in Deanna Kuhn's (2005) vision.


In the view of the scientific naturalists, knowledge of God was simply unattainable. Huxley, one of the most prominent spokesmen for this point of view, coined the term 'agnostic' to describe this conlusion."

p. 107.

Creationists nowadays attempt to revive the idealistic position of the early 19th century.


In traditional natural theology, God was known to exist because of the traces he had left behind. So, for example, the extraordinarily conplex and functional eye pointed directly to its cause: a designing God. The empiricists' strict liitation of knowledge made this kind of inference invalid. One could not assert the reality of the cause by knowledge of the effects in this case any more than in physics.

p. 106-107.

Projective or analytic geometry was my frustration in the last grades of the Gymnasium; somehow or other I could not grasp what it was or that it is a formalistic system, up until the day before being examined on the subject (and scoring the highest grade for it). The following quotation is rather explicit about what it is that is not understandable about projective geometry.

conceptual foundations?

Projective geometry, however, seemed to contain inconceivable elements. The points of intersection of two circles which neither touch nor cut each other have no place in the usual image of space. The point of intersection of a pair of parallel lines might be suggested by looking down a railroad track, but anyone who has traveled down such a track knows it to be an illusion. Imaginary points and points at infinity were not clearly conceivable. Yet these points were an essential part of projective geometry's success in providing geometrical generality to match that of analysis. In addition, they were essential to the Cayley/Klein theory of distance. The whole theory of the Absolute, which provided the additional points for the cross ration definition of distance, was composed of these evidently inconceivable points."

Chapter 3. p. 148.

This quotation bristles with problematic points for the educational philosopher, or the designer of achievement test items, for all that matters. I will not elaborate on them, however. The point is, teaching projective geometry without coming to grips with these unconceivabilities is a rather irresponsible act. Well then, let's do something about it. Other disciplines might also have their 'points at infinity' and imaginary intersections. Teaching them might leave the pupils in utter bewilderment.

In Richards' description of the public discussion about these things imaginary is quite amusing to see how the discovery of Neptune was used as an analogue: one day someone might succeed in giving an empirical interpretation to these imaginary points. Wishful thinking. Rather elaborate constructions of metrics (distance in projective space) proved to be circular. Not a success story, but how revealing the intellectual struggles are!

back to education

"In the context of discussions about teaching elementary geometry, England's mathematicians actively struggled with a number of practical problems which were involved in their descriptive view of mathematical truth. (...) Although the educational conflicts turned around elementary geometry, rather than the forefront of research where non-Euclidean and projective geometries were being developed, the basic questions being addressed were often strikingly similar."

p. 158.

Euclid, examinations, society

"The conservative defense of Euclid and his Elements was bolstered a great deal by the form of the prevailing institutional structure. In England, not only educational but often professional position and promotion in the army, navy, civil service and so on were routinely judged by standardized examinations. In geometry, these examinations assumed a thorough knowledge of specific theorems, proofs, orderings and so forth from Euclid's Elements. Rejecting Euclid as the basic text required drastically revising the examinations and developing a new, equally clear-cut and universal standard by which students could be judged."

p. 170.

Of course, the 1870 situation as here described by Richards is not really any different, structurally, from what prevails in the United States of America today, as regards admissions to higher education. There the SAT (Scholastic Aptitude Test) plays somewhat the role of Euclid's Elements and the first part of the post-1848 Mathematical Tripos at the same time. Try to remember this. More.

Richards does not seem to comment on the inconsistency in the conservative camp of promoting the value of Euclid's Elements for a humanistic education, and using the same book for sorting their students into different tracks in society. However, Richards does spend some space, I will quote some, on the derailments of the examinations, especially the mathematical Tripos, leaving it somewhat to the reader to conclude that the conservative camp managed to keep their ideals strictly separate from the daily routines they themselves as educators and examinators were involved in.

"Within a liberal education, many conservatives emphasized, it was not merely geometry which was being taught, it was Euclid's geometry. This gave elementary geometry not only a disciplinal, but a humanistic interest. [Whewell, 1845, Of a Liberal Education] When the students learned Euclid, they were initiated into a universe of discourse which had persisted for thousands of years. (...) However, the appeal of this kind of historical, humanistic argument was lost on the growing number of Englishmen who were pursuing geometry as preparation for life as an engineer or some other practical career."

p. 167-168.

John Perry's talk 1901: freedom at last

"What we want is a great Toleraton Act which will allow us all to pursue our own ideals, taking each from the other what he can in the way of mental help. We do not want to interfere with the students of pure mathematics, men whose peculiar mental processes are suited to these studies .... . The more they hold themselves in their studies as a race of demigods apart the better it may be for the world. ...
I belong to a great body of men who apply the principles of mathematcs in physical science and engineering; I belong to the very much greater body of men who may be called persons of average intelligence. In each of these capacities I need mental training and also mathematical knowledge."

John Perry (Ed.) BAAS Discussion on the teaching of Mathematics which took place in Glasgow, 1901, pp. 3-4. As quoted in Richards p. 197.

"His talk generated a three-hour discussion (...). One of the major themes running through the debate was the oppressiveness of educational structures which forced a single authority on all students and educators. (...)The practical effect of this discussion was virtually immediate. Again committees were formed in both the BAAS and the Mathematical Association to put pressure on the universities to modify their Euclidean-based examinations. In 1903 Cambridge agreed to accept 'any proof of the proposition, which appears to the Examiners to form part of a systematic treatment of the subject.' The other major universities quickly followed suit. The hegemony of Euclidean geometry in English education came to an abrupt end [Brock, Geometry, pp. 30-31] because the group for whom geometry was part of a prctical education finally broke the power of those who defined its value strictly in tems of liberal education.

p. 197-198.

Note that the examinations mentioned in the quotation must be the universities' admissions examinations, not the Mathematical Tripos of Cambridge. Of course, the Tripos was to change also.

"... Perry's aim was to free educators from the tight restrictions of the pure mathematicians. His 'toleration act' was designed to protect those who were interested in practical mathematics from the esoteric demands of rigor. The split he effected was equally liberating for the mathematicians, however.

p. 198.

stolen time

"The way geometry was tested in the late nineteenth-century English examination system was as confining for those who wanted to develop mathematics at its highest levels as it was for those trying to teach it to beginners.
This side of the educational picture was recognized by many of those who were pushing for reform. Even John Perry, in a mellow moment, noted that 'the greatest sufferer hitherto ... has been the real mathematician, who is drilled so long on elementary work that even after he becomes a wrangler [a winner of the highest honors in mathematics at Cambridge] he is only ready to begin that higher work which he might have studied years before.' [Perry (1907) 'The Mathematical Tripos at Cambridge,' Nature, 75, 274] Judging from retrospective reports, this seems to be an accurate representation of the situation for the mathematically inclined. Dissatisfaction with the mathematics which appeared on the Tripos is an almost universal theme in the memoirs of those who were educated late in the nineteenth century."

p. 201-202.

Not only late in the nineteenth century. The Tripos always has been ridiculed by its very wranglers. It is a prime example of assessment completely running out of hand, out of control, getting crazy, or whatever you may want to call it. There is absolutely no reason to romanticize the Tripos, be it the late nineteenth or the late eighteenth century ones.

Forsyte's frustration

"In patriotic duty bound, the Cambridge of Newton dhered to Newton's fluxions, to Newton's geometry, to the very text of Newton's Principia: in my own Tripos of 1881 we were expected to know any lemma in that great work by its number alone, as if it were one of the commandments or the 100th Psalm. Thus English mathematics were isolated: Cambridge became a school that was self-satisfied, self-supporting, self-content, almost marooned in its limitations."

A. R. Forsyth (1935). 'Old Tripos Days at Cambridge.' The Mathematical Gazette, 19, 167. As quoted in Richards, p. 202.

Russell recalls

"The mathematical teaching at Cambridge when I was an undergarduate was definitely bad ... . The necessity for nice discrimination between the abilities of different examinees led to an emphasis on 'problems' [memorized] as opposed to 'bookwork' [freeform]. The 'proofs' that were offered of mathematical theorems were an insult to the logical intelligence. Indeed, the whole subject of mathematics was presented as a series of clever tricks by which to pile up marks in the Tripos."

Bertrand Russell (1959). My Philosophical Development. George Allen & Unwin. pp. 37-38. As quoted in Richards, p. 202-203.

Karl Pearson expressed another opinion in (1936): 'Old Tripos Days at Cambridge, As Seen from Another Viewpoint. The Mathematical Gazette, 20, 27-36. Richards does not quote him, however.

the Tripos, an odd examination if ever there was one

"The spirit behind the movement for curricular reform at the university level was essentially the same as that which had grown up at the secondary level. A large part of the criticism that fueled the forces of change was directed against the structure of an education that pointed towards performance on a single examination. By the end of the century, that structure was undeniably odd. In order to do creditably on the Tripos, one needed to engage a coach, who would both teach the material which was likely to be tested and train the students in examination strategies. Often the latter concern seemed to overshadow the former, and the twists and turns of the Tripos, as opposed to relevant mathematics, became the focus of the students' attention."

p. 232-233.

At the beginning of that century that structure was odd already, of course.

how reliable ranking destroyed valid assessment

"This problem was exacerbated by the way the examination was marked. The results of the examination, which could be critically important to a student's future, were grouped into three major categories - Wrangler, First Class and Second Class - and students were rank listed within each category. The necessity of making distinctions among students which would be precise and fine enough to rank each of them individually meant that the examination was full of picky details, trick questions and memorized proofs. Thus students had to devote considerable time to exploring the obscure nooks and crannies of mathematical detail in order to prepare themselves for the Tripos competition. Furthermore, the examination contained more problems than anyone could possibly finish in the time allotted. This design was intended to give students choice. However, in practice, it meant that for the good students, the speed with which they wrote became the a major factor determining how well they placed. For this group, the examination could be as much a handwriting race as a test of learning. It is little wonder that many who learned mathematics within this system complained bitterly about it afterwards."

p. 233.

The above is a fine description of a paradigm case of ruining an assessment by unrestrained maximizing its reliability, something Borsboom, Mellenbergh and Van Heerde (2004, p. 1067) explicitly warned against. Whatever validity the assessment otherwise might have had, it was surely completely wasted by this desastrous policy.

Denny Borsboom, Gideon J. Mellenbergh and Jaap van Heerden, J. (2004). The concept of validity. Psychological Review, 111, 1061-1071. pdf

from ranking to grading

"At the end of the century, a number of reforms were proposed to deal with the educationally destructive idiosyncracies of the Tripos. The first, institued in 1892, allowed students to take the Tripos in the course of the second rather than the third year of their career. This was intended to free them to pursue their own interests in the final two years. In the event, however, this attempt to liberate students from the excessive rigors of tripos preparation was ineffective. The rankins were simply too important for students to jeopardize their standing with inadequate preparation.
Therefore an additional reform was proposed which was designed to mitigate the intense pressure to do well on the examination. This reform, seriously suggested in 1899 but not instituted until 1907, involved abolishing the order of merit; instead of establishing and publishing a strict rank ordering of all who had passed the examination, the results would be grouped alphabetically in three categories: Wrangler, Senior Optime and Junior Optime. By eliminating the need to make fine distinctions among candidates, the reform was designed to eliminate many of the excesses which abounded on the late-century Tripos (...).

p. 233-234.

The failure of the 1892 reform is a case of unanticipated washback-effects, or feed-forward effects. One might cynically remark that it was rather naive to expect much from this particular attempt to lessen the washback effects of the old regulation.

Abandoning the ranking itself was needed to alleviate the exaggerated forms of washback of the Tripos. This is one of those moments in the history of assessment where explicitly a change from ranking to the use of modern grades is effected. For anothr such change, in several steps and in France, see my (1998) html.

I must mention also that in this her last chapter Joan Richards explains the importance of the 1892 change for the place of mathematics in education, and for mathematics itself. The liberal educational ideal was thrown from its throne.

the new middle class, and Germany's power

"... the arguments for taking the Tripos early were often couched in terms of its advantage to those who wished to use mathematics in pursuit of other special areas. (...). The practical needs of engineers or physicists were a major focus of interest. (...) It also reflects the particular situation at the turn of the century. At that time England was gripped in a xenophobic reaction to Germany's growing strength" (p. 236).

p. 236.


David E. Joyce (www). Euclid's Elements. html

On my website

Ben Wilbrink (1997). Assessment in historical perspective. Studies in Educational Evaluation, 23, 31-48. html

My matheducation page

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